THE OTHER HIGGSES, AT RESONANCE, IN THE LEE- WICK EXTENSION OF THE - PowerPoint PPT Presentation

THE OTHER HIGGSES, AT RESONANCE, IN THE LEE- WICK EXTENSION OF THE STANDARD MODEL ARXIV:1108.3765, JHEP10 (2011) 145 (IN COLLABORATION WITH ROMAN ZWICKY) Dr. Terrance Figy The University of Manchester Birmingham Particle Physics Seminars 29 Feb 2012

OUTLINE • The Lee-Wick Standard Model • Higgs boson pair production • Top quark pair production • Conclusions

LEE-WICK STANDARD MODEL (LWSM) B.Grinstein, D.O’Connel, M.B.Wise (2007) Based on ideas by Lee and Wick (1969,1970 )

A TOY MODEL B. Grinstein, D. O’Connel, M.B. Wise (2007) (A) HD formalism: L hd = 1 1 φ ) 2 − 1 φ 2 − 1 φ∂ µ ˆ 2 M 2 ( ∂ 2 ˆ 2 m 2 ˆ 2 ∂ µ ˆ 3! g ˆ φ 3 , φ − Propagator: ˆ D ( p ) = i ( p 2 − p 4 /M 2 − m 2 ) − 1 2 poles: p 2 = m 2 , M 2 (B) AF formalism: ˆ φ = φ − ˜ φ L = 1 2 ∂ µ φ∂ µ φ − 1 φ + 1 φ 2 − 1 φ ) 2 − 1 φ∂ µ ˜ 2 M 2 ˜ 2 ∂ µ ˜ 2 m 2 ( φ − ˜ 3! g ( φ − ˜ φ ) 3 . Wrong sign kinetic and mass term M. The two formulations are equivalent. Use EoM.

A TOY MODEL B. Grinstein, D. O’Connel, M.B. Wise (2007) ˜ φ φ φ φ φ φ + i − i ˜ D ( p ) = ; D ( p ) = p 2 − m 2 p 2 − M 2 d 4 p d 4 p i i � � Σ (0) = ig p 2 − m 2 − ig (2 π ) 4 (2 π ) 4 p 2 − M 2 d 4 p i ( m 2 − M 2 ) � = ig (2 π ) 4 ( p 2 − m 2 )( p 2 − M 2 ) Quadratic divergence is cancelled leading to a logarithmic divergence.

A TOY MODEL B. Grinstein, D. O’Connel, M.B. Wise (2007) − i − i − i − i Σ ( p 2 ) � � φ ( p ) = p 2 − M 2 + p 2 − M 2 + . . . D ˜ p 2 − M 2 − i = p 2 − M 2 + Σ ( p 2 ) . � g 2 1 − 4 m 2 − i φ ( p ) = Γ = D ˜ p 2 − M 2 − iM Γ , M 2 . 32 π M A LW resonance has a probability of decaying in the interval . Γ dt − dt Is this a problem? Shall we debate this issue further or proceed?

LWSM: SUMMARY • For each SM field add a higher derivative (HD) term. • Auxiliary fields (AF) can be introduced to cast the theory in terms of interactions with mass dimension no greater than 4. • The AFs are interpreted as LW partner states and have the wrong-sign propagator (aka Pauli-Villars regulators). • The LWSM solves the hierarchy problem: the extra minus sign in the loop diagrams come from the LW field propagators. No need for opposite spin statistics! • Unitarity is preserved, provided that the LW fields do no appear as asymptotic states in the S-matrix. • Causality is preserved at the the macroscopic level (where we live). However, there can be violations of causality at the microscopic level.

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LWSM Higgs Sector (AF formalism) D µ ˜ H † ˜ L = ( ˆ D µ H ) † ( ˆ D µ H ) � ( ˆ D µ ˜ H ) † ( ˆ H ˜ H � V ( H � ˜ H ) + M 2 H ) , L µ T a + g 2 W a µ T a + g 1 B µ Y h A µ = gA a where ˆ D µ = ∂ µ + i ( A µ + ˜ A µ ) w ˜ gauge the two doublets are p p H > = H > = ˜ ⇥ ˜ h + , (˜ ⇥ ⇤ ⇤ 0 , ( v + h 0 ) / 2 , h 0 + i ˜ p 0 ) / 2 h ˜ h h 0 i = v , h 0 i = 0 . h 0 ) 2 + M 2 L mass = � λ 4 v 2 ( h 0 � ˜ 2 (˜ h 0 ˜ p 0 + 2˜ h + ˜ H h 0 + ˜ p 0 ˜ h � ) . mixing between the Higgs scalar and its LW–partner. Th

LWSM Higgs Sector ! ! ! h cosh φ h sinh φ h h phys Symplectic rotation: = ˜ ˜ h sinh φ h cosh φ h h phys Mass eigenvalues: ˜ h 0 h 0 p 0 ˜ h ± CP even even odd none m 2 q q ⇣ ⌘ ⇣ ⌘ 1 1 1 � 2 v 2 λ /M 2 1 � 2 v 2 λ /M 2 phys 1 � 1 + 1 1 M 2 H H 2 2 H

LWSM Higgs Sector Mixing angle: 2 m 2 r h 0 ≡ m h 0 , phys λ v 2 = h 0 , phys h 0 ) , , (1 + r 2 m ˜ h 0 , phys 1 s H = cosh φ h = h 0 ) 1 / 2 , (1 − r 4 1 + r 2 h 0 s H � ˜ H = cosh φ h − sinh φ h = h 0 ) 1 / 2 . (1 − r 4

LWSM Yukawa Interactions (in auxiliary field formalism) L = Ψ t i η 3 ˆ D Ψ t − Ψ t R M t η 3 Ψ t L η 3 M † Ψ t L − Ψ t / R , L , ˜ t R , ˜ Ψ t > Ψ t > L = ( T L , ˜ t 0 R = ( t R , ˜ T 0 T L ) , R ) SU(2) doublet: g. Q L = ( T L , B L ) > s which in turn form 0 1 0 1 m t 0 − m t 1 0 0 M t η 3 = A , η 3 = − m t − M u m t 0 − 1 0 B C B C @ @ A 0 0 − M Q 0 0 − 1

LWSM Diagonalization of mass matrices Ψ L ( R ) , phys = η 3 S † M t, phys η 3 = S † L ( R ) η 3 Ψ L ( R ) , R M t η 3 S L , S L η 3 S † S R η 3 S † L = η 3 and R = η 3 Higgs-quark vertices L = − 1 − 1 ⇣ ⌘ ⇣ ⌘ v ( h 0 − ˜ L g † L g † R g t Ψ t t Ψ t R g t Ψ t t Ψ t Ψ t L + Ψ t Ψ t L − Ψ t h 0 ) v ( − i ˜ p 0 ) R R 0 1 m t 0 − m t g t, phys = S † g t = A , R g t S L − m t 0 m t B C @ 0 0 0

LWSM LW gauge bosons are massive and mix: 1 B µ ν + W µ ν W µ ν − ˜ B µ ν B µ ν − ˜ � W µ ν � B µ ν ˜ W µ ν ˜ L 2 g = − 2Tr a ) + g 2 2 v 2 1 2 ˜ B µ + M 2 2 ˜ B µ ˜ µ ˜ ( W 1 , 2 + ˜ W 1 , 2 µ ) 2 W a W µ 2( M 1 − µ 8 v 2 B µ + g 2 W 3 W 3 µ ) 2 8 ( g 1 B µ + g 1 ˜ µ + g 2 ˜ + Gauge interactions: [ g 1 ¯ ψ ( � B + � ˜ B ) ψ + g 2 ¯ ψ ( � W + � ˜ � L int = W ) ψ ] − ψ = qL,uR,dR g 1 ¯ ψ + g 2 ¯ � � ψ ( � B + � ˜ ˜ B )˜ ψ ( � W + � ˜ ˜ W ) ˜ � + ψ . ψ = q,u,d

LWSM Couplings to gauges bosons and fermions E. Alvarez, E. Coluccio, J.Zurita: arXiv 1004.3496 t F ˜ 1 / 2 ( β t P σ ( gg → ˜ 1 + r 2 g ˜ P ) P ) Pt ¯ | 2 , ˜ g h 0 f ¯ f = − g ˜ f = cosh θ − sinh θ = 1 − r 4 , g ˜ f = − 1 . g 2 √ Pgg = σ SM ( gg → H ) = | h 0 f ¯ Pf ¯ ˜ F 1 / 2 ( β t P ) ˜